The developers still aren't sure what your computer will need, but Sims 3 will have options to put your graphics on lower resolution just like Sims 2. So while you might need a better computer to play with high-resolution, I wouldn't stress too much yet about being able to play the game at all. The developers are very concerned with making the game playable on older machines.
Consider it from a mercenary standpoint. The more people with capable computers, the more copies of the game they'll sell. And with how many times they've said "It'll run on your grandma's PC!" this is a very big issue to them. They are not going to let it be a huge monster that only state-of-the-art computers can run.
So You Still Want to Improve Your System?
If you want to improve your computer without buying a new one, you just need to update your ram and graphics card (aka video card). If you don't know how to install these things and don't know anyone who can help, don't buy anything on your own, just take it to a place you trust to install a new one. National chains will gouge you on the price, but they'll also get the job done.
If it's your processor that's outdated, it's cheapest to get a new computer due to compatability issues.
Minimum system requirements will play the game at the lowest settings, recommended will play it at normal, but if you want the best performance, you need a better system than they ask for. CanYouRunIt.com lets you choose your game from a list and download an ActiveX/Java application which examines your computer and gives you an analysis, including recommendations for upgrades if they're needed. You can uninstall the application when you're finished.
(Everything here is a generalization and can vary from game to game, system to system, and part to part.)
